I don't think mammoths would have any problems surviving today.
They could not bring back any parasites, they are just taking dna. That said, most of the parasites we have in the world today are extremely ancient and have been around a long, long time.
I think the mammoth would be pretty comfortable hanging out with elephants. They are similar enough. The mammoth is basically an elephant that was able to manage in cold weather.
And I think what they eat is fairly similar too, based on them having same teeth. They eat very low quality forage.
I think from its surrogate mother, the mammoth would learn to react to humans much as the elephants do. Basically, a nice animal, likes people, but stay out of the pen when it's irritated.
They would probably do very well in large parts of the world. The trick would be creating a new herd in which all the members are not too closely related. DNA would need to be taken from animals from different parts of the world.
As for 'not messing with mother nature'...O.M.G. Man has done nothing but mess with mother nature, cutting down whole forests, changing entire continents. Bringing back the mammoth would be not even on the list compared to what all else has been done.